Minimum H-decompositions of graphs

Given graphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a graph isomorphic to H. Let ϕ H ( n ) be the smallest number ϕ such that any graph G of order n admits an H-decomposition with at most ϕ parts. e determine the asymptotic of ϕ H ( n ) for any fixed graph H as n tends to infinity. act computation of ϕ H ( n ) for an arbitrary H is still an open problem. Bollobás [B. Bollobás, On complete subgraphs of different orders, Math. Proc. Cambridge Philos. Soc. 79 (1976) 19–24] accomplished this task for cliques. When H is bipartite, we determine ϕ H ( n ) with a constant additive error and provide an algorithm returning the exact value with running time polynomial in log n .